Notes

1A complete listing of all general and special face forms of the 32 point groups is given in ch. 10 of Hahn & Klapper (2002BB21), Table 10.1.2.2; the eigensymmetries of the 47 face forms are listed in Table 10.1.2.3. Illustrations of all face forms are contained in ch. 3.2 (pp. 184-188) of the book by Vainshtein (1994BB45), ch. 3 (Fig. 73) of Shubnikov & Koptsik (1974BB40) and ch. 10 of Buerger (1956BB7).

2Representative twin operation m'(1[{\overline 1}]0).

3Note that this illustration does not require crystals to exhibit planar (habit) faces and complete face forms. It is applicable to crystals of any (also spherical) shape to be studied by X-ray diffraction. This is shown in Section 2.3, p. 336, of Klapper & Hahn (2010BB29).

4In some cases of reticular merohedral twins certain indices are always integer, e.g. for the tetragonal [Sigma]5 and hexagonal [Sigma]7 twins which preserve the tetragonal or hexagonal axis. Here the index l is always integer (cf. §§5[link] and 6[link]).

5The rhombohedral [Sigma]3 twins treated here are `twins by reticular merohedry with parallel threefold axes'. They are thus independent of the c/a ratio (hexagonal axes) or the rhombohedral angle [alpha] (rhombohedral axes) and can occur in any rhombohedral crystal. Twins `with inclined threefold axes' depend on the axial ratio or on [alpha]. Both types have been derived by Grimmer (1989bBB18).

6They are the same as those in Table 9, subtable (c) of Klapper & Hahn (2010BB29) for crystals with hexagonal lattices and [Sigma]1 twin laws 2[001] and m(0001). This is due to the fact that face forms are independent of the lattice type (hexagonal or rhombohedral).

7In principle, however, domains of [Sigma]3 obverse/reverse reflection twins with twin laws 3 [rightwards arrow] 3m, 3 [rightwards arrow] [\overline 6 \equiv 3/m] and 32 [rightwards arrow] [\overline 6 2m] can be studied by the reversal of their optical rotation sense. The same applies to the corresponding trigonal [Sigma]1 twins.

8Exact lattice coincidence is, in principle, not possible because the coincidence is not enforced by symmetry as it is in the case of `parallel c-axis' twins. It may occur, however, for a certain temperature if the thermal expansion is anisotropic.

9Q = (k1 - 2P)2 + P2 = (h1 - P)2/4 + P2 for coincidence condition h1 + 2k1 = 5P; similar for h2, k2 with h2 + 2k2 = 5P.

10The d values of reflections hkl of a tetragonal crystal are given by 1/d2 = (h2 + k2)/a2 + l2/c2. Since l2/c2 is the same for both twin-related reflections, the d values are equal for equal h2 + k2.

11Q = 7P2 + h1(h1 - 5P) = [7P2 + k1(k1 + 4P)]/4 for coincidence condition 2h1 - k1 = 7P. The same holds for h2, k2 and condition 2h2 - k2 = 7P.

12The d values of reflections hkl of a hexagonal crystal are given by 1/d2 = (h2 + hk + k2)/a2 + l2/c2. Since l2/c2 is the same for both twin-related reflections, the d values are equal for equal h2 + hk + k2.

13The reduced composite symmetries are the same for 3m1 and 31m etc., because the twin elements m'(12[\overline 3]0) etc. are the same for both groups. For details of `structural settings', see Klapper & Hahn (2010BB29), Appendix A.

14Note that there are four conjugate subgroups [\overline 3]2/m of 4/m[\overline 3]2/m of index [4]. Each of these has a single [\overline 3] axis along one of the four body-diagonal [\overline 3] axes of the cubic supergroup (cf. Müller, 2004BB34, p. 708).

15If the subgroup is chosen along one of the other body diagonals [[\overline 1]11], [1[\overline 1]1] or [11[\overline 1]], the types of splitting, given below, remain the same, but the Miller indices of the subforms would change.

16They represent the four conjugate subgroups [\overline 3]2/m of 4/m[\overline 3]2/m.

17It is emphasized that the term `opposite' (i.e. related by an inversion) refers here only to the morphology of the two split forms.